Radar pulse compression repair

ABSTRACT

A radar pulse compression repair (RPCR) system includes a receiver for receiving a radar return signal, a matched filter for applying matched filtering to the radar return signal to generate a matched filter output, a processor programmed for applying Radar Pulse Compression Repair (RPCR) to the matched filter output to suppress a plurality of range sidelobes from the matched filter output, and a detector for receiving the RPCR-processed output. The RPCR invention in operating upon the output of the matched filter enables RPCR to be employed as a post-processing stage in systems where it is not feasible to replace the existing pulse compression apparatus. RCPR can also be selectively employed when it is possible that large targets are present that may be masking smaller targets, thereby keeping computational complexity to a minimum.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application claims the benefit of the priority filing dateof provisional patent application No. 60/626,502, filed Nov. 8, 2004,incorporated herein by reference.

BACKGROUND OF THE INVENTION

The present invention relates to radar signal processing. Moreparticularly, the present invention relates to the post-matched filterprocessing repair of radar return signals.

Range sidelobe suppression in radar pulse compression has been a topicof intense scrutiny for several years. The goal is to minimize theeffects caused by the correlation of a transmitted waveform with delayedversions of itself while maintaining close to the optimalsignal-to-noise ratio (SNR). One approach has sought to achieve this byemploying some form of least squares (LS) estimation; see, for example,U.S. Pat. No. 5,805,107. LS has been shown to provide the most efficientestimator when the additive noise is white. However, LS is not robust inthat it does not account for scatterers closer than some nominal rangeR₀, which can have a deleterious effect on the estimation of the channelat the ranges of interest.

Another approach termed Reiterative Minimum Mean-Square Error (RMMSE)estimation and described in U.S. Pat. No. 6,940,450, issued Sep. 6, 2005and incorporated herein by reference, provides a robust estimate of theradar channel impulse response that is almost completely devoid of rangesidelobes. This approach, however, as with LS estimation, involvesreplacing the standard matched filter within legacy radar systems, whichis not always a feasible solution.

It would therefore be desirable to provide an approach for decreasingthe radar sidelobes after a,standard matched filter radar signalprocessing operation.

BRIEF SUMMARY OF THE INVENTION

According to the invention, a radar pulse compression repair (RPCR)system includes a receiver for receiving a radar return signal, amatched filter for applying matched filtering to the radar return signalto generate a matched filter output, a processor programmed for applyingRadar Pulse Compression Repair (RPCR) to the matched filter output tosuppress a plurality of range sidelobes from the matched filter output,and a detector for receiving the RPCR-processed output. RPCR ispreferably repeated until the range sidelobes are suppressed to thelevel of a noise floor. In a preferred embodiment, the received radarreturn signal at the l^(th) range gate is defined asy(l)=x ^(T)(l)s+v(l)   (1)for l=0, . . . ,L+N−2, where x(l)=[x(l) x(l−1) . . . x(l−N+1)]^(T) is aset of impulse response coefficients that a transmitted waveform s isconvolved with at delay l, v(l) is additive noise, (•)^(T) is atranspose operation, and L is a number of range gates in a processingwindow, a system response model based on collecting N samples of thereceived radar return signal is expressed asy(l)=[x(l)x(l+1) . . . x(l+N−1)]^(T) s+v(l)=X ^(T)(l)s+v(l)   (2)where y(l)=y[y(l)+1) . . . y(l+N−1)]^(T) and v(l)=[v(l) v(l+1) . . .v(l+N−1)]^(T), the matched filtering operation comprises convolving thereceived radar return signal with a time-reversed complex conjugate ofthe transmitted waveform as expressed in the digital domain as{circumflex over (x)} _(MF)(l)=s ^(H) y(l)   (3),the convolution of the transmitted waveform with the radar impulseresponse (1) and the convolution of the received radar return signalwith the time-reversed, complex conjugated waveform (3) are combined andrepresented as{circumflex over (x)} _(MF)(l)=r ^(T) {tilde over (x)}( l)+u(l)   (4),where {tilde over (x)}(l)=[x(l+N−1) . . . x(l+1) x(l) x(l−1) . . .x(l−N+1)]^(T), u(l) is an additive noise correlated by the matchedfiltering, and r is a length 2N−1 result of the convolution of thetransmitted waveform s and a receive filter {overscore (s)}*=└s*(N−1)s*(N−2) . . . s*(0)┘, collecting 2N−1 samples of the matched filteroutput (4) and representing is as the received radar return signal as inequation (1) to which the filtering operation{circumflex over (x)} _(RPCR)(l)=w(l){tilde over ({circumflex over(x)})}_(MF)(l)   (5)is performed in which an assumed transmitted waveform is r, estimatingan optimal receive filter for each individual range gate asw(l)=ρ(l) (C(l)+R)⁻¹ r   (6)where ρ(l)=E[|x(l)|²] is an expected power of x(l), or alternativelyρ(l)=E[|x(l)|^(α)], where values for α preferably fall within 1≦α≦1.7,R=E[u(l) u^(H)(l)] is a noise covariance matrix, and the matrix C(l) isdefined as $\begin{matrix}{{C(l)} = {\sum\limits_{n = {{{- 2}N} + 2}}^{{2N} - 2}{\rho\quad\left( {l - n} \right)r_{n}r_{n}^{H}}}} & (7)\end{matrix}$in which r_(n) contains the elements of the length 2N−1 waveformautocorrelation r right-shifted by n samples and the remainder zerofilled; (6) may be applied reiteratively to improve the accuracy of theestimate. In the embodiment in which the invention employs a stabilityfactor, α replaces the exponent in ρ(l)=E[|x(l)|²] resulting inρ(l)=E[|x(l)|α],where values for α preferably fall within 1≦α≦1.7, in the processingalgorithm to keep the matrix C(l) from becoming ill-conditioned when thereceived radar return signal has a large dynamic range.

The RPCR invention advantageously operates upon the output of thematched filter. This therefore enables RPCR to be employed as apost-processing stage in systems where it is not feasible to replace theexisting pulse compression apparatus. RCPR can also be selectivelyemployed when it is possible that large targets are present that may bemasking smaller targets, thereby keeping computational complexity to aminimum.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an RCPR system according to the invention.

FIG. 2 is a schematic diagram of the RCPR processing methodologyaccording to the invention.

FIG. 3 is a graph showing range sidelobes resulting from matchedfiltering and the repair of the matched filter output using RPCR for onelarge target and one small target in noise according to the invention.

FIG. 4 is a graph showing a matched filter output and the repair of thematched filter output using RPCR for dense target scenario.

DETAILED DESCRIPTION OF THE INVENTION

Definitions: The term “convolution” means the process that yields theoutput response of an input to a linear time-invariant system, such asis described and defined in J. G. Proakis and D. G. Manolakis, DigitalSignal Processing:Principles, Algorithms, and Applications, 3rd Ed., pp.75-82, Prentice Hall: Upper Saddle River, N.J. (1996), incorporatedherein by reference. The term “deconvolution” as used herein means theprocess that given the output of a system determines an unknown inputsignal to the system. See Id. at p. 355, incorporated herein byreference. The term “scatterer” means something in the path of atransmitted waveform that causes a significant reflection (relative tothe noise) back to the receiver of the sensor.

The new radar pulse compression repair (RPCR) method and system is basedon applying Reiterative Minimum Mean-Square Error (MMSE) estimation(described further below) after standard pulse compression via matchedfiltering.

Referring now to FIG. 1, an RPCR system 10 includes a radar transmitter12 for transmitting a radar pulse waveform 14 through a transmissionenvironment 15 toward a target (not illustrated) resulting in a radarreturn signal 16.

We denote the discrete-time version of the transmitted waveform 14 asthe column vector s having length N. A receiver 18 receives a radarreturn signal 16. According to the range resolution of the transmittedwaveform, the received signal 16 at the l^(th) range gate is defined asy(l)=x ^(T)(l)s+v(l)   (1)

for l=0, . . . , L+N−2, where x(l)=[x(l) x(l−1) . . . x(l−N+1)]^(T) isthe set of impulse response coefficients representing the radarscattering objects in the environment which reflected the transmittedwaveform 14 s at delay l, v(l) is additive noise, (•)^(T) is thetranspose operation, and L is the number of range gates in theprocessing window. Collecting N samples of the received radar returnsignal 16, the system response model is expressed asy(l)=[x(l)x(l+1) . . . x(l+N−1)]^(T) s+v(l)=X ^(T)(l)s+v(l)   (2)

where y(l)=[y(l) y(l+1) . . . y(l+N−1)]^(T) and v(l)=[v(l) v(l+1) . . .v(l+N−1)]^(T). A matched filter 20 convolves the received radar returnsignal 16 with the time-reversed complex conjugate of the transmittedwaveform 14 which can be expressed in the digital domain as{circumflex over (x)} _(MF)(l)=s ^(H) y(l).   (3)

Following matched filtering, a legacy radar system may employ an A/Dconverter 22 as well-as some additional processing 24 prior to applyinga detector 26 to the matched filter output. However, it is well-knownthat the matched filter 20 generates range sidelobes in range cells nearto large target returns. An example of this is depicted in FIG. 3 forthe matched filter output using a length N=30 Frank-Kretschmer P3 code(waveform) whereby the range sidelobes from the large target mask thesmall nearby target. For targets that greatly exceed the level of thenoise (by 60 dB in the example but typically by 30+ dB depending on theparticular waveform) , smaller targets can be masked by the residualrange sidelobes in the range cells surrounding large targets.

In some legacy radar systems it is not feasible to replace the existingpulse compression apparatus to enable robust range sidelobe suppression.However, range sidelobe suppression can still be achieved bypost-processing the output of matched filter 20 via an analog-to-digital(AD) convertor 22 and a processor 24 for applying RPCR to the digitizedmatched filter output prior to the detector 24. The received signalmodel for RPCR involves the combination of the operations of convolutionof the transmitted waveform 14 with the radar impulse response (i.e.equation (1)) and the convolution of the received return signal 16 withthe time-reversed, complex conjugated waveform via the matched filter 20(i.e. equation (3) which is represented as{circumflex over (x)} _(MF)(l)=r ^(T) {tilde over (x)}(l)+u(l)   (4)

where {tilde over (x)}(l)=[x(l+N−1) . . . x(l+1) x(l) x(l−1) . . .x(l−N+1)]^(T), u(l) is additive noise correlated by the matched filter,and r is the length 2N−1 convolution of the transmitted waveform s andthe receive filter {overscore (s)}*=└s*(N−1) s*(N−2) . . . s*(0)┘ (inother words r is the autocorrelation of s) Treating the matched filteroutput in (4) as the received return signal (i.e. as in equation (1)), avariant of the RMMSE adaptive pulse compression algorithm described inU.S. Pat. No. 6,940,450, issued Sep. 6, 2005 and incorporated herein byreference, termed Radar Pulse Compression Repair (RPCR) , is applied tothe matched filter output as{circumflex over (x)}_(RPCR)(l)=w ^(H)(l){tilde over ({circumflex over(x)})}_(MF)(l)   (5)

in which the assumed transmitted waveform for the RPCR received signalmodel is r.

The RPCR algorithm estimates the optimal receive filter for eachindividual range gate asw(l)=ρ(l) (C(l)+R)⁻¹ r   (6)where ρ(t)=E[|x(l)|²] is the expected power of x(l), R=E└u(l) u^(H)(l)┘is the noise covariance matrix, and the matrix C(l) is defined as$\begin{matrix}{{C(l)} = {\sum\limits_{n = {{{- 2}N} + 2}}^{{2N} - 2}{\rho\quad\left( {l - n} \right)r_{n}r_{n}^{H}}}} & (7)\end{matrix}$

in which r_(n) contains the elements of the length 2N−1 waveformautocorrelation r right-shifted by n samples and the remainder zerofilled.

Given the range cell estimates from the output of the matched filter,one can then reiteratively apply (6) in order to further improve theaccuracy of the estimate. It has been found that one to three stagesallows RPCR to robustly suppress the range sidelobes to the level of thenoise floor when the radar return channel is somewhat sparselyparameterized (as is the case with high range resolution radii).

Note that each reiteration stage will reduce the number of range gateestimates by 2(2N−2). To counteract this, it is preferred to increasethe window of range gates by 2M(2N−2), where M is the number of stages.

Finally, to maximize numerical efficiency, RPCR may be implemented usinga variation of the matrix inversion lemma.

In a preferred embodiment, a stability factor α replaces the exponent inρ(l)=E[|x(l)|²] in (6) resulting inρ(l)−E[|x(l)|^(α)]  (8)

The stability factor is used to keep the matrix C(l) from becomingill-conditioned when the received radar return signal has a largedynamic range. Preferred values for α fall within 1≦α≦1.7. Furthermore,similar to the adaptation step-size in closed-loop algorithms, it ispreferable to set α at the high end initially and allow it to decreaseto the low end by the final stage.

EXAMPLES

As an example, we use an N=30 Lewis-Kretschmer P3 code (waveform) for aradar impulse response containing one large target (with 60 dB SNR) anda smaller target (with 20 dB SNR). The RPCR algorithm is applied to thematched filter output using 1 stage with the stability parameter set asα=1.6. FIG. 3 presents the ground truth and matched filter output aswell as the results of the RPCR algorithm.

The single stage of the RPCR algorithm results in a reduction of theMean-Square Error (MSE) between the estimate and ground truth of 34 dBover the MSE of the matched filter. Also, the mismatch filter response(the matched filter response divided by the RPCR response) for the rangecell containing the large target is only 0.05 dB.

For a more stressing example, we examine the performance when there arenumerous targets: both large and small. We employ 2 stages of the RPCRalgorithm with α₁=1.6 and α₂=1.4. In FIG. 4 we see that many smalltargets are masked by the larger targets in the matched filter output.By comparison, RPCR is able to suppress the range sidelobes that resultfrom the matched filter down to the level of the noise. In so doing, italso reveals many smaller targets that were masked by larger neighboringtargets and almost exactly coincides with the ground truth. In terms ofthe MSE, the RPCR reduces the MSE exhibited by the matched filter by 41dB.

Obviously many modifications and variations of the present invention arepossible in light of the above teachings. It is therefore to beunderstood that the scope of the invention should be determined byreferring to the following appended claims.

1. A method for post-processing repair of a radar return signal,comprising: a) receiving a radar return signal; b) applying matchedfiltering to said radar return signal to generate a matched filteroutput; and c) applying Radar Pulse Compression Repair (RPCR) to saidmatched filter output to thereby suppress a plurality of range sidelobesfrom said matched filter output.
 2. A method as in claim 1, wherein RPCRis repeated until the range sidelobes are suppressed to the level of anoise floor.
 3. A method as in claim 1, wherein the received radarreturn signal at the l^(th) range gate is defined asy(l)=x ^(T)(l)s+v(l)   (1) for l=0, . . . , L+N−2, where x(l)=[x(l)x(l−1) . . . x(l−N+1)]^(T) is a set of impulse response coefficientsthat a transmitted waveform s is convolved with at delay l, v(l) isadditive noise, (•)^(T) is a transpose operation, and L is a number ofrange gates in a processing window, a system response model based oncollecting N samples of the received radar return signal is expressed asy(l)=[x(l)x(l+1) . . . x(l+N−1)]^(T) s+v(l)=X ^(T)(l)s+v(l)   (2) wherey(l)=[y(l) y(l+1) . . . y(l+N−1)]^(T) and v(l)=[v(l) v(l+1) . . .v(l+N−1)]^(T), the matched filtering operation comprises convolving thereceived radar return signal with a time-reversed complex conjugate ofthe transmitted waveform as expressed in the digital domain as{circumflex over (x)} _(MF)(l)=s ^(H) y(l)   (3), the convolution of thetransmitted waveform with the radar impulse response (1) and theconvolution of the received radar return signal with the time-reversed,complex conjugated waveform (3) are combined and represented as{circumflex over (x)} _(MF)(l)=r ^(T) {tilde over (x)}( l)+u(l)   (4),where {tilde over (x)}(l)=[x(l+N1) . . . x(l+1) x(l) x(l−1) . . .x(l−N+1)]^(T), u(l) is an additive noise correlated by the matchedfiltering, and r is a length 2N−1 convolution of the transmittedwaveform s and a receive filter {overscore (s)}*=└s*(N−1) s*(N−2) . . .s*(0)┘, substituting the matched filter output (4) as the received radarreturn signal in equation (1) for the operation{circumflex over (x)} _(RPCR)(l)=w ^(H)(l){circumflex over ({tilde over(x)})} _(MF)(l)   (5) in which an assumed transmitted waveform is r,estimating an optimal receive filter for each individual range gate asw(l)=ρ(l) (C(l)+R)⁻¹ r   (6) where ρ(l)=E[|x(l)|²] is an expected powerof x(l), R=E└u(l) u^(H)(l)┘] is a noise covariance matrix, and thematrix C(l) is defined as $\begin{matrix}{{C(l)} = {\sum\limits_{n = {{{- 2}N} + 2}}^{{2N} - 2}{\rho\quad\left( {l - n} \right)r_{n}r_{n}^{H}}}} & (7)\end{matrix}$ in which r_(n) contains the elements of the length 2N−1waveform autocorrelation r right-shifted by n samples and the remainderzero filled.
 4. A method as in claim 3, wherein (6) is appliedreiteratively to improve the accuracy of the estimate.
 5. A method as inclaim 4, wherein a window of range gates is increased by 2M(2N−2) whereM is the number of reiteration stages.
 6. A method as in claim 1,wherein the received radar return signal at the l^(th) range gate isdefined asy(l)=x ^(T)(l)s+v(l)   (1) for l=0, . . . , L+N−2, where x(l)=[x(l)x(l−1) . . . x(l−N+1)]^(T) is a set of impulse response coefficientsthat a transmitted waveform s is convolved with at delay l, v(l) isadditive noise, (•)^(T) is a transpose operation, and L is a number ofrange gates-in a processing window, a system response model based oncollecting N samples of the received radar return signal is expressed asy(l)=[x(l)x(l+1) . . . x(l+N−1)]^(T) s+v(l)=X ^(T)(l)s+v(l)   (2) wherey(l)=[y(l) y(l+1) . . . y(l+N−1)]^(T) and v(l)=[v(l) v(l+1) . . .v(l+N−1)]^(T), the matched filtering operation comprises convolving thereceived radar return signal with a time-reversed complex conjugate ofthe transmitted waveform as expressed in the digital domain as{circumflex over (x)} _(MF)(l)=s ^(H) y(l)   (3), the convolution of thetransmitted waveform with the radar impulse response (1) and theconvolution of the received radar return signal with the time-reversed,complex conjugated waveform (3) are combined and represented as{circumflex over (x)} _(MF)(l)=r ^(T) {tilde over (x)}( l)+u(l)   (4),where {circumflex over (x)}(l)=[x(l+N−1) . . . x(l+1) x(l) x(l−1) . . .x(l−N+1)]^(T), u(l) is an additive noise correlated by the matchedfiltering, and r is a length 2N−1 convolution of the transmittedwaveform s and a receive filter {overscore (s)}*=└s*(N−1) s*(N−2) . . .s*(0)┘, substituting the matched filter output (4) as the received radarreturn signal in equation (1) for the operation{circumflex over (x)} _(RPCR)(l)=w ^(H)(l){tilde over ({circumflex over(x)})} _(MF)(l)   (5) in which an assumed transmitted waveform is r,estimating an optimal receive filter for each individual range gate asw(l)=ρ(l) (C(l)+R)⁻¹ r   (6) where ρ(l)=E[|x(l)^(α)] is an expectedpower of x(l), R=E└u(l) u^(H)(l)┘ is a noise covariance matrix, and thematrix C(l) is defined as $\begin{matrix}{{C(l)} = {\sum\limits_{n = {{{- 2}N} + 2}}^{{2N} - 2}{\rho\quad\left( {l - n} \right)r_{n}r_{n}^{H}}}} & (7)\end{matrix}$ in which r_(n) contains the elements of the length 2N−1waveform autocorrelation r right-shifted by n samples and the remainderzero filled.
 7. A method as in claim 6, wherein 1≦α≦1.7.
 8. A method asin claim 6, wherein (6) is applied reiteratively to improve the accuracyof the estimate.
 9. A method as in claim 8, wherein a window of rangegates is increased by 2M(2N−2) where M is the number of reiterationstages.
 10. A method for post-processing repair of a radar returnsignal, comprising: receiving a radar return signal, wherein saidreceived radar return signal at an l^(th) range gate is defined asy(l)=x ^(T)(l)s+v(l)   (1) for l=0, . . . , L+N−2, where x(l)=[x(l)x(l−1) . . . x(l−N+1)]^(T) is a set of impulse response coefficientsthat a transmitted waveform s is convolved with at delay l, v(l) isadditive noise, (•)^(T) is a transpose operation, and L is a number ofrange gates in a processing window; expressing a system response modelbased on collecting N samples of the received radar return signal asy(l)=[x(l)x(l+1) . . . x(l+N−1)]^(T) s+v(l)=x ^(T)(l)s+v(l)   (2) wherey(l)=[y(l) y(l+1) . . . y(l+N−1)]^(T) and v(l)=[v(l) v(l+1) . . .v(l+N−1)]^(T); applying matched filtering to said received radar returnsignal to generate a matched filter output, wherein said matchedfiltering operation comprises convolving-the received radar returnsignal with a time-reversed complex conjugate of the transmittedwaveform as expressed in a digital domain as{circumflex over (x)} _(MF)(l)=s ^(H) y(l)   (3), combining theconvolution of the transmitted waveform with the radar impulse response(1) and the convolution of the received radar return signal with thetime-reversed, complex conjugated waveform (3) such that{circumflex over (x)} _(MF)(l)=r ^(T) {tilde over (x)}( l)+u(l)   (4),where {tilde over (x)}(l)=[x(l+N−1) . . . x(l+1) x(l) x(l−1) . . .x(l−N+1)]^(T), u(l) is an additive noise correlated by the matchedfiltering, and r is a length 2N−1 convolution of the transmittedwaveform s and a receive filter {overscore (s)}*=└s*(N−1) s*(N−2) . . .s*(0)┘, substituting the matched filter output (4) as the received radarreturn signal in equation (1) for the operation{circumflex over (x)} _(RPCR)(l)=w ^(H)(l){tilde over ({circumflex over(x)})} _(MF)(l)   (5) in which an assumed transmitted waveform is r,estimating an optimal receive filter for each individual range gate asw(l)=ρ(l) (C(l)+R)⁻¹ r   (6) where ρ(l)=E[|x(l)|^(α)] is an expectedpower of x(l), R=E└u(l) u^(H)(l)┘ is a noise covariance matrix, and thematrix C(l) is defined as $\begin{matrix}{{C(l)} = {\sum\limits_{n = {{{- 2}N} + 2}}^{{2N} - 2}{\rho\quad\left( {l - n} \right)r_{n}r_{n}^{H}}}} & (7)\end{matrix}$ in which r_(n) contains the elements of the length 2N−1waveform autocorrelation r right-shifted by n samples and the remainderzero filled.
 11. A method as in claim 10, wherein (6) is appliedreiteratively to improve the accuracy of the estimate.
 12. A method asin claim 10, wherein 1≦α≦1.7.
 13. A method as in claim 10, wherein awindow of range gates is increased by 2M(2N−2) where M is the number ofreiteration stages.
 14. A radar pulse compression repair (RPCR) system,comprising: a receiver for receiving a radar return signal; a matchedfilter for applying matched filtering to said radar return signal togenerate a matched filter output; a processor, wherein said processor isprogrammed for applying Radar Pulse Compression Repair (RPCR) to saidmatched filter output to thereby suppress a plurality of range sidelobesfrom said matched filter output and thereby provide an RPCR-processedoutput; and a detector for receiving the RPCR-processed output.
 15. Asystem as in claim 14, wherein RPCR is repeated until the rangesidelobes are suppressed to the level of a noise floor.
 16. A system asin claim 14, wherein the received radar return signal at the l^(th)range gate is defined asy(l)=x ^(T)(l)s+v(l)   (1) for l=0, . . . , L+N−2, where x(l)=[x(l)x(l−1) . . . x(l−N+1)]^(T) is a set of impulse response coefficientsthat a transmitted waveform s is convolved with at delay l, v(l) isadditive noise, (•)^(T) is a transpose operation, and L is a number ofrange gates in a processing window, a system response model based oncollecting N samples of the received radar return signal is expressed asy(l)=[x(l)x(l+1) . . . x(l+N−1)]^(T) s+v(l)=X ^(T)(l)s+v(l)   (2) wherey(l)=[y(l) y(l+1) . . . y(l+N−1)]^(T) and v(l)=[v(l) v(l+1) . . .v(l+N−1)]^(T), the matched filtering operation comprises convolving thereceived radar return signal with a time-reversed complex conjugate ofthe transmitted waveform as expressed in the digital domain as{circumflex over (x)} _(MF)(l)=s ^(H) y(l)   (3), the convolution of thetransmitted waveform with the radar impulse response (1) and theconvolution of the received radar return signal with the time-reversed,complex conjugated waveform (3) are combined and represented as{circumflex over (x)} _(MF)(l)=r ^(T) {tilde over (x)}( l)+u(l)   (4),where {tilde over (x)}(l)=[x(l+N−1) . . . x(l+1) x(l) x(l−1) . . .x(l−N+1)]^(T), u(l) is an additive noise correlated by the matchedfiltering, and r is a length 2N−1 convolution of the transmittedwaveform s and a receive filter {overscore (s)}*=└s*(N−1) s*(N−2) . . .s*(0)┘, substituting the matched filter output (4) as the received radarreturn signal in equation (1) for the operation{circumflex over (x)} _(RPCR)(l)=w ^(H)(l){tilde over ({circumflex over(x)})} _(MF)(l)   (5) in which an assumed transmitted waveform is r,estimating an optimal receive filter for each individual range gate asw(l)=ρ(l) (C(l)+R)⁻¹ r   (6) where ρ(l)=E[|x(l)|²] is an expected powerof x(l), R=E└u(l) u^(H)(l)┘ is a noise covariance matrix, and the matrixC(l) is defined as $\begin{matrix}{{C(l)} = {\sum\limits_{n = {{{- 2}N} + 2}}^{{2N} - 2}{\rho\quad\left( {l - n} \right)r_{n}r_{n}^{H}}}} & (7)\end{matrix}$ in which r, contains the elements of the length 2N−1waveform autocorrelation r right-shifted by n samples and the remainderzero filled.
 17. A system as in claim 16, wherein (6) is appliedreiteratively to improve the accuracy of the estimate.
 18. A system asin claim 17, wherein a window of range gates is increased by 2M(2N−2)where M is the number of reiteration stages.
 19. A system as in claim14, wherein the received radar return signal at the l^(th) range gate isdefined asy(l)=x ^(T)(l)s+v(l)   (1) for l=0, L+N−2, where x(l)=[x(l) x(l−1) . . .x(l−N+1)]^(T) is a set of impulse response coefficients that atransmitted waveform s is convolved with at delay l, v(l) is additivenoise, (•)^(T) is a transpose operation, and L is a number of rangegates in a processing window, a system response model based oncollecting N samples of the received radar return signal is expressed asy(l)=[x(l)x(l+1) . . . x(l+N−1)]^(T) s+v(l)=X ^(T)(l)s+v(l)   (2) wherey(l)=[y(l) y(l+1) . . . y(l+N−1)]^(T) and v(l)=[v(l) v(l+1) . . .v(l+N−1)]^(T), the matched filtering operation comprises convolving thereceived radar return signal with a time-reversed complex conjugate ofthe transmitted waveform as expressed in the digital domain as{circumflex over (x)} _(MF)(l)=s ^(H) y(l)   (3), the convolution of thetransmitted waveform with the radar impulse response (1) and theconvolution of the received radar return signal with the time-reversed,complex conjugated waveform (3) are combined and represented as{circumflex over (x)} _(MF)(l)=r ^(T) {tilde over (x)}( l)+u(l)   (4),where {tilde over (x)}(l)=[x(l+N−1) . . . x(l+1) x(l) x(l−1) . . .x(l−N+1)]^(T), u(l) is an additive noise correlated by the matchedfiltering, and r is a length 2N−1 convolution of the transmittedwaveform s and a receive filter {overscore (s)}*=└s*(N−1) s*(N−2) . . .s*(0), substituting the matched filter output (4) as the received radarreturn signal in equation (1) for the operation{circumflex over (x)} _(RPCR)(l)=w ^(H)(l){tilde over ({circumflex over(x)})} _(MF)(l)   (5) in which an assumed transmitted waveform is r,estimating an optimal receive filter for each individual range gate asi w(l)=ρ(l) (C(l)+R)⁻¹ r   (6) where ρ(l)=E[|x(l)|^(α)] is an expectedpower of x(l), R=E└u(l) u^(H)(l)┘ is a noise covariance matrix, and thematrix C(l) is defined as $\begin{matrix}{{C(l)} = {\sum\limits_{n = {{{- 2}N} + 2}}^{{2N} - 2}{\rho\quad\left( {l - n} \right)r_{n}r_{n}^{H}}}} & (7)\end{matrix}$ in which r_(n) contains the elements of the length 2N−1waveform autocorrelation r right-shifted by n samples and the remainderzero filled.
 20. A system as in claim 19, wherein 1≦α≦1.7.
 21. A systemas in claim 19, wherein (6) is applied reiteratively to improve theaccuracy of the estimate.
 22. A system as in claim 14, wherein a windowof range gates is increased by 2M(2N−2) where M is the number ofreiteration stages.
 23. A system as in claim 9, further comprising ananalog-to-digital converter for receiving and. converting the matchedfilter output to a digital signal for processing by RPCR.